Analytic LO Gluon Distributions from the proton structure function F 2 (x,Q 2 )---

Jan., 2010 Aspen Winter Physics Conference XXVI M. Block

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Analytic LO Gluon Distributions from the proton structure function F2(x,Q2)---

---> New PDF's for the LHCMartin Block

Northwestern University Happy 25th

Anniversary, Aspen Winter Conferences


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“Analytic derivation of the leading-order gluon distribution function G(x,Q2)=xg(x,Q2) from the proton structure function F2p(x,Q2)”, M. M. Block, L. Durand and D. McKay, Phys. Rev. D 77, 094003 (2008).

Outline of talk


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“Analytic treatment of leading-order parton evolution equations: Theory and tests”, M. M. Block, L. Durand and D. McKay, Phys. Rev. D 79, 04031 (2009).

“A new numerical method for obtaining gluon distribution functions G(x,Q2)=xg(x), from the proton structure function”,M. M. Block, Eur. Phys. J. C. 65, 1 (2010).


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“Small-x behavior of parton distributions from the observed Froissart energy dependence of the deep-inelastic-scattering cross sections”, M. M. Block, Edmund L. Berger and Chung-I Tan, Phys.Rev. Lett. 308 (2006).


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Doug Randy Phuoc Ha

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TEAM GLUON

Fellow authors and collaborators:Fellow authors and collaborators: to be blamed!


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F2 is the proton structure function, measured by ZEUS at HERA


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This talk concentrates exclusively on

extracting an analytical solution G(x,Q2) of the

DGLAP evolution equation involving F2 for

LO or Fs for NLO


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Same F2 as for DIS scheme, or LO MSbar

F20 and G are convoluted with NLO MSbar

coefficient functions Cq and Cg

We solve this NLO convolution equation for F20(x,Q2) directly by

means of Laplace transforms, so that we find F20(x,Q2) as a function

of F2p(x,Q2).


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This illustrates the case for nf = 4; depending on Q2, we also use nf = 3 and 5


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For LO, it’s simpler: the proton structure function

F2(x,Q2) --> G(x,Q2) directly, with NO approximations

We also need s(Q2)


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Simple for LO, and don’t depend on Q2


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Same general form of equations for both

LO and NLO


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The convolution theorem for Laplace transforms


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Not enough time for details of inversion algorithm: See M. M. Block, Eur. Phys. J. C. 65, 1 (2010).


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Blue dots = GMSTW

Red Curves = Numerical Inversion of Laplace transform

NLO GMSTW2008, Q2 = 1, 5, 20, 100, Mz2 GeV2,


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LO G(v), using ZEUS data, from Laplace Numerical Inversion of g(s), for Q2 = 5 GeV2, where v = ln(1/x)

Blue Dots = Exact Analytic SolutionRed Curve= numerical inversion of Laplace transform.Derived from global fit to ZEUS F2(x,Q2), Fig.1, M. M. Block, EPJC. 65, 1 (2010).


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Results of an 8-parameter fit to ZEUS proton structure function data for x<0.09. The renormalized d.f. =1.1


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LO Gluon Distributions:

GCTEQ6L compared to our ZEUS LO G(x), for Q2 = 5, 20 and 100 GeV2

CTEQ6L

Kinematic HERA

boundary

Why are there large differences where there are F2 data?


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Look at Proton structure functions, F2 , compared to ZEUS data:

1) CTEQ6L, constructed from LO quark distributions,

2) Our fit to ZEUS data, Q2 = 4.5, 22 and 90 GeV2

CTEQ6L

CTEQ6L disagrees with experimental ZEUS data!


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Proton structure functions, F2 , compared to ZEUS data:

1) MSTW2008, constructed from NLO quark distributions,

2) Our fit to ZEUS data , Q2 = 4.5, 22 and 90 GeV2

NLO MSTW

MSTW2008 does much better than CTEQ6L, but still not a good fit


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NLO G(x) , constructed from a fit to ZEUS F2 data, compared to MSTW2008, for Q2 = 100 and Mz

2 GeV2

Dashed = our G Solid = NLO MSTW

Very different gluon values at the Z mass

Note the different shapes for G derived from F2 data compared to G from evolution---a remnant of MSTW assuming parton distribution shapes at Q0

2 = 1 GeV2. Differences grow larger as Q2 increases!


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LO and NLO G(x) , from MSTW2008, for Q2 = 10, 30 and 100 GeV2

Dashed = NLO

Solid = LO

Enormous differences between gluon

distributions for small x, for next order in s ; no large changes in quark

distributions


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Dashed = NLO

Solid = LO

LO and NLO G(x) , from F2 fit to ZEUS, for Q2 = 10, 30 and 100 GeV2

Again, very large differences between

gluon distributions for small x, for next order in s ; what does LO gluon

mean?


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Conclusions

1. We have shown that detailed knowledge of the proton structure function F2(x,Q2) and s(Q2) determines G(x)=xg(x); for LO, it is all that is necessary. For NLO, addition of tiny terms involving NLO partons are required for high accuracy.

2. No a priori theoretical knowledge or guessing of the shape of the gluon distribution at Q0

2---where evolution starts--- is needed; experimental measurements determine the shape!

3. Our gluon distributions at small x disagree with both LO CTEQ6L and NLO MSTW2008, even in regions where there are structure function F2 data.


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4. We think that the discrepancies are due to both CTEQ , MSTW assuming shape distributions at Q0

2 that are wrong; remnants of the assumed shape are retained at high Q2, through the evolution process. This effect becomes exacerbated at small x!

5. Message! Don’t trust “standard candles” at LHC.

Future

PLEA! Make publicly available combined ZEUS and H1 structure function data (with correlated errors) so that we can make more accurate gluon distributions using the combined HERA results.

Incorporate mass effects in splitting functions, to avoid discontinuities near c and b thresholds.

Analytic LO Gluon Distributions from the proton structure function F 2 (x,Q 2 )---

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